A variant of Mathias forcing that preserves \(\mathsf{ACA}_0\) (Q1938403)

This paper formulates and applies \(F_\sigma\)-Mathias forcing, an adaptation of set-theoretic Mathias forcing specially devised to preserve properties of models of second-order arithmetic. For example, the author proves that if \(\mathcal N\) is a model of ACA\(_0\) and \(G\) is generic for \(F_\sigma\)-Mathias forcing, then \(\mathcal N [G]\) is also a model of ACA\(_0\). This statement also holds with ACA\(_0\) replaced by WKL\(_0\) plus induction for \(\Sigma^0_2\) formulas. The final section demonstrates the use of this forcing machinery in obtaining conservation and cone-avoidance theorems. The article supplies a conceptual bridge from the set-theoretic treatment of Mathias forcing of \textit{J. E. Baumgartner} [Lond. Math. Soc. Lect. Note Ser. 87, 1--59 (1983; Zbl 0524.03040)] and \textit{A. Blass} [Ann. Pure Appl. Logic 109, No. 1--2, 77--88 (2001; Zbl 0980.03055)] to the computability-theoretic results of \textit{P. A. Cholak} et al. [J. Symb. Log. 66, No. 1, 1--55 (2001; Zbl 0977.03033); corrigendum ibid. 74, No. 4, 1438--1439 (2009; Zbl 1182.03107)] and \textit{D. D. Dzhafarov} and \textit{C. G. Jockusch jun.} [J. Symb. Log. 74, No. 2, 557--578 (2009; Zbl 1166.03021)].